Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, t\neq 0$. $\dfrac{{(a^{-1}t^{4})^{-1}}}{{(at^{2})^{5}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{-1}t^{4})^{-1} = (a^{-1})^{-1}(t^{4})^{-1}}$ On the left, we have ${a^{-1}}$ to the exponent ${-1}$ . Now ${-1 \times -1 = 1}$ , so ${(a^{-1})^{-1} = a}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{-1}t^{4})^{-1}}}{{(at^{2})^{5}}} = \dfrac{{at^{-4}}}{{a^{5}t^{10}}}$ Break up the equation by variable and simplify. $\dfrac{{at^{-4}}}{{a^{5}t^{10}}} = \dfrac{{a}}{{a^{5}}} \cdot \dfrac{{t^{-4}}}{{t^{10}}} = a^{{1} - {5}} \cdot t^{{-4} - {10}} = a^{-4}t^{-14}$